Based on my experience with my kids, I'd start with a .22LR rifle, preferably single-shot or bolt action. Then they can move on to a .22lr semi-auto. When they get to the point where they won't just be missing more expensively, a mild centerfire will definitely be of interest. I made mine show they could put 10/10 in an 8" target at 25 yards before they got the expensive ammo. Not a real hard test, but it focused them a little. Good next steps would be M1 Carbine or AR in 5.56/.223. I think they have about the same recoil. I have a video clip of my boy shooting an m1 carbine for the first time... "Woah!" He dug it so much.
At any jump in caliber I start them with a single round for a couple of reps, especially in a semi-auto.
Pistols are fascinating to most kids, but few can hit anything with them. A narrow grip will fit their smaller hands. I found my CZ-Kadet, which is a full sized pistol with a grip large enough for a double stack mag, was too big for my 10 year old. A 1911 fits better. I think bringing the target *way* in, to 10-15 feet, is probably right for pistols. If you aren't on paper, you probably won't learn much from your target.
Since you were asking about calibers and not what I've been talking about, back to the boy: He shot and loved a Ruger GP100 with .357 ammo. I'm not sure what the specs on that ammo were. He did not love the 1911 with 230 grain bullets. Pretty sure the GP100 is a lot heavier and soaks up recoil better.
My 16 year old daughter likes .45acp. .30-06 intimidates her, just not her favorite. I've seen a clip of a 10-11 year old girl at Appleseed hitting a gong at 200 yards with a Garand, and liking it. Her form was great, though. I only saw two shots, so she might have developed a flinch later. I'd probably keep it to a couple of rounds per outing from a big boomer. Maybe the last two in the clip so she gets the "ping!" of the clip ejecting. My daughter also shoots her boyfriend's Mosin once in a while, a couple of shots every few outings. She'll shoot an AR all day.
I'm sure there are kids out there shooting 12 gauge slugs all day, too. I think introducing a heavier caliber a step at a time will let them find the right level of challenge where the recoil is interesting but not overwhelming. This means you have to have a good selection of guns!
It's a lot more fun with kids. Sometimes a bit stressful, I know I don't get to shoot much with all the supervision and support. The malfunction drill is "lay it on the table and get Daddy" at this point. We talk squibs and hang fire vs misfire, but I'm not sure they really *get* it yet. So I handle things while talking them through my process. "You had a 'click', but no 'boom'. Ok, it's been about 15 seconds, so let's eject the round carefully - watch my muzzle discipline and how I keep my face away from the chamber - and see if the bullet is still seated in the case. It is? Good, no squib. Did we get a light strike or did the firing pin leave a deep impression...."
Just my rambling thoughts when I'm supposed to be working. Take care! Safe and happy shooting, all.